Liouville theorem for elliptic equations with mixed boundary value conditions and finite Morse indices

In this paper, we establish Liouville type theorem for boundedness solutions with finite Morse index of the following mixed boundary value problems: − Δ u = | u | p − 1 u in R + N , ∂ u ∂ ν = | u | q − 1 u on Γ 1 , ∂ u ∂ ν = 0 on Γ 0 , and − Δ u = | u | p − 1 u in R + N , ∂ u ∂ ν = | u | q − 1 u on Γ 1 , u = 0 on Γ 0 , where R + N = { x ∈ R N : x N > 0 } , Γ 1 = { x ∈ R N : x N = 0 , x 1 < 0 } and Γ 0 = { x ∈ R N : x N = 0 , x 1 > 0 } . The exponents p , q satisfy the conditions in Theorem 1.1 .